You may want to see if there are any engine codes. I was thinking of a speed sensor as well. Also, you didn't say what led you to change the battery. If you think it was running down, the low voltage may have contributed or even caused the problems you saw. If the new battery is also giving out, your problem could be your alternator. One failure mode of the alternator is that diodes that are supposed to allow current to charge the battery can fail allowing current to leak back from the battery through the alternator coils and discharge the battery. When a new battery dies quickly, the next thing to replace is the alternator.
So I guess you have a couple things to look at: speed sensor and alternator. Reading the codes may point to one or the other. Also, if you take the alternator out of the car, many parts stores can check the alternator performance on a bench tester that will check how will it's charging and whether the diodes are working properly. | 
